KANNE, Circuit Judge.
Larry Collins was convicted of conspiracy to distribute methamphetamine and now raises four claims on appeal. He alleges that the district court erred by permitting the government to bolster the credibility of its witnesses and by giving both a factually incorrect jury instruction and an untimely “dynamite” jury instruction. He also challenges the constitutionality of the mandatory life sentence for multiple drug-crime convictions. We find that the factually incorrect jury instruction was error, but it was harmless under the circumstances. We affirm Collins’s conviction.
I. History
A federal grand jury issued a three-count superseding indictment against Collins, including one count of conspiracy to distribute and possess with intent to distribute methamphetamine in violation of 21 U.S.C. §§ 841 and 846, and two counts of money laundering in violation of 18 U.S.C. § 1956. The indictment alleged that, from January 1993 to June 1996, Collins “knowingly and intentionally conspired with Warren Wegman, Robin Allen, David Ma-bry, Earl Jackson Hiland III and with others to possess with the intent to distribute and to distribute a controlled substance, namely methamphetamine.”
Allen, Mabry and Hiland each testified against Collins at his trial pursuant to plea agreements in connection with guilty pleas on unrelated drug charges. In its opening statement, the government announced that it would present the testimony of Collins’s co-conspirators and introduced the trio’s plea agreements into evidence over Collins’s objections. All but one of the plea agreements emphasized that the witness had agreed to provide “complete and truthful” testimony, and each witness averred that he or she was obligated by plea agreement to be truthful.
Robin Allen testified that she began selling methamphetamine in 1992, when Weg-man began delivering money and buying from her on Collins’s behalf. Allen testified that Collins continued to purchase methamphetamine from her the following year, using Wegman as a runner. David Mabry, Allen’s boyfriend at the time, joined her during the summer of 1993, and the couple became Collins’s primary drug source. Allen and Mabry testified that between June 1993 and May 1996, they supplied methamphetamine to Collins about ten times in two- to five-pound batches. In addition to those sales, Allen and Mabry earned a million dollars between 1993 and 1996 by serving a host of customers other than Collins, including Dale Daugherty.
Daugherty operated his own drug-dealing operation, which had no connection with Collins. In fact, Collins instructed Allen not to deal with his competitor Daugherty because it “would take food off my table.” Allen agreed to cease working with Daugherty, but Mabry continued selling to Daugherty through a third party without her help. After their arrests in May 1996, Allen and Mabry both pleaded guilty to participation in a conspiracy headed by Daugherty to distribute methamphetamine. However, by the government’s admission both at trial and on appeal, the Daugherty conspiracy did not involve Collins, and the conspiracy for which Collins was charged constituted a separate and distinct operation. The government adduced telephone records that listed calls between Daugherty and Allen to corroborate Allen’s and Mabry’s testimony, but the government was careful to explain that Daugherty was not a member of Collins’s operation. The government declared, “we’re not alleging that Mr. Daugherty ... was a part of the conspiracy in which the defendant was charged.”
The third co-conspirator to testify against Collins was Earl Hiland. He testified that he traveled to California in 1993 or 1994 to acquire methamphetamine from Allen for Collins and visited St. Louis, Missouri, in 1994 to obtain methamphetamine from an Allen affiliate. Hiland was arrested in May 1996 and pleaded guilty to possession with intent to distribute. At trial, Hiland admitted that “there was absolutely no involvement of Larry Collins” with his arrest and the charges to which he pleaded guilty “didn’t have anything to do with Larry Collins.” Collins’s attorney challenged Hiland’s truthfulness on cross-examination, just as he had attacked the credibility of Allen and Mabry during their testimony. The government responded to each challenge by highlighting the fact that the witnesses were obligated under plea agreement to tell the truth.
The government also presented several other witnesses who had either worked
with Collins or purchased drugs from him. Michael Peters, a friend of Wegman, testified that he purchased methamphetamine from Collins in 1992, but shortly thereafter began buying directly from Allen by mail. In 1994, Peters was arrested and pleaded guilty to possession of methamphetamine purchased from Allen. Peters testified at trial that his arrest and the underlying charges had nothing to do with Collins. During closing argument, the government based its case on the wealth of testimony against Collins, particularly that of Allen, Mabry and Hiland. The government again stressed that these witnesses were testifying pursuant to plea agreements and were bound to tell the truth. The government emphasized that there was “powerful motive for each one of these witnesses to tell you the truth.”
At the close of trial, Collins objected to Jury Instruction No. 14, a cautionary instruction reciting the following:
You have heard testimony from Earl Jackson Hiland III, Herman Hudson, Robin Allen, David Mabry, Frank Ciko-vich, Cheri Knowles, Ola Redes, Sonny Logan, Michael Peters and Belinda Peters, who received immunity; that is a promise from the government that any testimony or other information they provided would not be used against them in a criminal case.
Earl Jackson Hiland III, Robin Allen, David Mabry and Michael Peters have stated that they were involved in the alleged conspiracy charged against the defendant and that each has pled guilty to a crime arising out of the same occurrence for which the defendant is now on trial.
Further, David Mabry, Carolyn Burd, Arduth Sapp and Belinda Peters have admitted lying under oath, and David Mabry has been convicted of lying under oath.
You may give the testimony of these witnesses such weight as you feel it deserves, keeping in mind that it must be considered with caution and great care. Moreover, the guilty pleas of Earl Jackson Hiland III, Robin Allen, David Ma-bry and Michael Peters are not to be considered as evidence against the defendant.
Evidence that a witness has been convicted of a crime is to be considered by you only insofar as it may affect the witness’s credibility.
Collins argued that the second paragraph of this jury instruction was incorrect because Allen and Mabry had pleaded guilty to charges pertaining to the Daugherty conspiracy, not the Collins operation, and Hiland and Peters had pleaded guilty to charges that each testified had nothing to do with Collins. Unpersuaded, the district court gave the instruction over Collins’s objection.
After a day of deliberation, the jury announced that it was “deadlocked” with respect to the conspiracy count. The court convened a conference that day with all counsel present to determine how to proceed. Collins’s attorney acknowledged that giving a “dynamite charge” would violate
United States v. Silvern,
484 F.2d 879 (7th Cir.1973), because the court had not issued the instruction to the jury in its initial charge. Collins’s attorney, however, consulted "with Collins and authorized the court to give Seventh Circuit pattern instruction 7.06, a modification of the
Silvern
deadlock instruction. He explained to the government and the court:
I believe that the law is and still is in the Seventh Circuit that it’s error to give that instruction after the jury has been instructed if it wasn’t tendered originally. We would waive any objection to the giving of that instruction. I’ve discussed this with Mr. Collins and we would certainly put our waiver on the record.
We certainly would like to have the jury return a verdict in the case and we would suggest that the instruction be given to the jury.... And we are cer
tainly willing to put a waiver on the record to any error caused by giving of the instruction by the timing.
The court then asked the following questions to Collins directly:
Court: I would ask Mr. Collins, directly, if he concurs with what his counsel has just stated?
Collins: Yes, sir.
Court: And do you agree and waive any problem that might exist as far as a legal consequence is concerned?
Collins: Yes, sir.
Court: And do you wish the Court at this time to give the 7.06 instruction, is that right?
Collins: Yes, sir.
Out of an “abundance of caution,” the district court also asked Collins’s attorney to explain for the record Collins’s reasons for waiving his objection. Collins’s attorney cited the attorney-client privilege but reiterated, “I discussed with Mr. Collins and read with him the content of the instruction and discussed with him the consequences of giving and not giving the instruction so that he could reach a reasoned judgment on that.” The court again asked Collins if he agreed with his attorney, and Collins again confirmed his consent, “Yes, sir.” The court concluded, “[Tjhat’s as far as we can probably go.... [T]hat is their decision and I accept that.”
The court then read the
Silvern
instruction to the jury,
and later that day, the jury convicted Collins on the conspiracy charge. On September 30, 1998, the court sentenced Collins, who had two prior drug convictions, to mandatory life imprisonment and ten years supervised release, pursuant to the “three-strikes” sentencing provision of 21 U.S.C. § 841(b)(l)(A)(viii).
II. Analysis
Collins raises four issues on appeal. First, Collins claims that the district court gave a jury instruction that misstated the testimony of Allen, Mabry, Hiland and Peters. Second, Collins argues that the district court committed reversible error by giving the
Silvern
instruction without having previously included it in the initial jury charge. Third, Collins contends that the district court improperly allowed the government to bolster the credibility of its witnesses. Fourth, Collins challenges the constitutionality of his mandatory life sentence.
A. Jury Instruction No. II
Jury Instruction No. 14 served a cautionary purpose, expressing “the idea that testimony from witnesses who have been paid for assistance should be received with greater skepticism (‘greater care’ or ‘caution and great care’ or ‘more caution’) than the testimony of other witnesses.... Lower punishment is one coin in which witnesses may be paid.”
United States v. Cook,
102 F.3d 249, 251 (7th Cir.1996). The intent of the instruction was to warn the jury that certain government wit
nesses, including Allen, Mabry, Hiland and Peters, had received leniency in exchange for their testimony and that their questionable credibility was subject to the jury’s careful consideration. We review for abuse of discretion whether the jury instructions, “in the light most favorable to the government” and “viewed as a whole, address the issues fairly and adequately.”
United States v. Stillo,
57 F.3d 553, 559 (7th Cir.1995). Even if the instruction was erroneous, Collins must establish actual prejudice to win reversal.
See United States v. Aldaco,
201 F.3d 979, 990 (7th Cir.2000).
Collins challenges the accuracy of the second paragraph in the instruction, which states that “Earl Jackson Hiland III, Robin Allen, David Mabry and Michael Peters have stated that they were involved in the alleged conspiracy charged against the defendant and that each has pled guilty to a crime arising out of the same occurrence for which the defendant is now on trial.” Allen and Mabry pleaded guilty to crimes connected to the Daugherty conspiracy, which the government agreed was unrelated to Collins’s operation. The Daugherty conspiracy therefore is not “the alleged conspiracy charged against the defendant,” nor did those crimes underlying the Daugherty conspiracy “aris[e] out of the same occurrence” as the charges against Collins. Furthermore, Hiland and Peters both unequivocally stated that they were arrested for and pleaded guilty to crimes having no connection to Collins. This directly contradicts the instruction’s assertion that they “pleaded guilty to a crime arising out of the same occurrence” as Collins’s crimes. Indeed, Peters never testified that he had any involvement in the Collins operation and testified only that he had purchased methamphetamine from Collins in 1992, two years before his own arrest. Jury Instruction No. 14 was factually incorrect.
Nonetheless, when viewed in the light most favorable to the government, the district court’s error here was harmless, though narrowly so. As Collins acknowledges, the trial record is clear that Collins was not involved with the Daugherty conspiracy. Collins’s indictment alleges that Collins conspired with Wegman, Allen, Mabry and Hiland, but neither the indictment nor the jury instructions mention Daugherty even once. At trial, the witnesses distinguished the Collins and Daugherty drug-dealing operations and characterized Collins and Daugherty as competitors. Allen testified that Collins, her grandfather, instructed her “[n]ot to do anything with Dale,” and Mabry confirmed that Collins had told Allen that “dealing with Dale Daugherty would take food off [Collins’s] table.” Peters testified that Wegman and Collins had quarreled over money and Wegman “was looking for the new money man.” As a result, Weg-man stopped dealing with Collins, and “[Wegman and Daugherty] got together and started making their own trips.” Similarly, another member of the Daugherty conspiracy, Ola Redes, testified that she had worked for Daugherty but that Collins was not involved and she had never seen Collins with any methamphetamine. Outside the presence of the jury, Collins’s attorney agreed that Wegman’s and Ma-bry’s testimony established Daugherty as “a competitor of the defendant” and stated that “[he did not] think there is anything to connect Mr. Daugherty in any conspiracy with [Collins].” The government agreed that it was “not alleging that Mr. Daugherty ... was part of the conspiracy in which the defendant was charged.” Despite the erroneous jury instruction, it is unlikely that the jury would confuse the Collins and Daugherty conspiracies or believe that Collins participated in the criminal activity of the Daugherty operation.
When viewed in this context, the jury instructions were not so misleading that the jury would likely be confused or misled by the district court’s error. Jury Instruction No. 14 was merely a cautionary instruction directed to the credibility of the witnesses, not to Collins’s guilt or culpabil
ity. Although the instruction was inaccurate, the thrust of the instruction was that the prosecution witnesses had cut deals with the government and possessed incentives to incriminate Collins; the instruction did not pose as a direction on Collins’s involvement in criminal activity. In addition, we assess jury instructions “when considered in their entirety and not in isolation,”
Reed v. Union Pacific R.R. Co.,
185 F.3d 712, 715 (7th Cir.1999), and the other instructions make clear that the jury was required to evaluate Collins’s guilt independently from the guilt of the testifying co-conspirators. Jury Instruction No. 14 itself commands that “the guilty pleas of Earl Jackson, Hiland III, Robin Allen, David Mabry and Michael Peters are not to be considered as evidence against the defendant.” Jury Instruction No. 21 directed, “In determining whether the defendant became a member of the conspiracy you may consider only the evidence concerning the acts and 'statements of that particular defendant.” Although the instruction was factually inaccurate, the trial sufficiently established that the two men were competitors rather than collaborators, and as a result, we find that the jury was not improperly influenced by the inaccurate cautionary instruction.
For the same reason, the faulty jury instruction did not constructively amend Collins’s indictment or create an improper variance between the indictment and proof. To establish either claim, Collins must show that he was convicted based on proof running to the Daugherty conspiracy, rather than the conspiracy for which he was indicted.
See United States v. Willoughby,
27 F.3d 263, 265-66 (7th Cir.1994). As we have explained, we believe that the jury was not confused by the incorrect association in Instruction No. 14 between the Daugherty and Collins conspiracies. The trial established that the two operations were independent and that Collins had no involvement with the Daugherty conspiracy to which Allen and Mabry had pleaded guilty. The government provided sufficient evidence relating to the Collins conspiracy that “a reasonable trier of fact could have found beyond a reasonable doubt the existence of the single conspiracy charged in the indictment.”
United States v. Townsend,
924 F.2d 1385, 1389 (7th Cir.1991). Allen and Mabry testified that they dealt extensively with Collins in distributing methamphetamine but explained that Collins and Daugherty were competitors. Hiland testified that he purchased methamphetamine from Collins and also made at least two trips to buy methamphetamine for Collins from suppliers. Lavon Logan testified that Collins provided him methamphetamine, which Logan then distributed on the street. This testimony, in addition to testimony from other witnesses, co-conspirator statements and corroborating phone records, provides sufficient evidence to support Collins’s conviction for knowingly and intentionally conspiring to distribute a controlled substance with Wegman, Allen, Mabry and Hiland, as his indictment charges.
B.
Silvern
Instruction
In
Allen v. United States,
164 U.S. 492, 501, 17 S.Ct. 154, 41 L.Ed. 528 (1896), the Supreme Court upheld a trial court’s issuance to a deadlocked jury of a “dynamite” supplemental jury instruction, which encouraged the jurors to reconsider their individual positions and asked minority dissenters whether they “might not reasonably doubt the correctness of a judgment which was not concurred in by the majority.” Subsequent to
Allen,
lower courts and commentators vigorously debated whether the efficiency gains of avoiding mistrials outweigh the risks of prejudice and jury coercion flowing from the use of dynamite charges.
See, e.g., United States v. Johnson,
432 F.2d 626, 630-34 (D.C.Cir.1970);
United States v. Fioravanti,
412 F.2d 407, 414-20 (3d Cir.1969); Note,
Due Process, Judicial Economy and the Hung Jury: A Re-Examination of the Allen Charge,
53 Va. L.Rev. 123 (1967); Note,
Deadlocked Juries and Dynamite: A Crit
ical Look at the
“Allen
Charge,”
31 U. Chi. L.Rev. 386 (1964). Faced with this controversy in
United States v. Silvern,
484 F.2d 879, 883 (7th Cir.1973), we established a clear procedure for the issuance of supplemental jury instructions to deadlocked juries by providing a model dynamite charge and explaining that it may be given to the jury after deliberations reach a deadlock, provided that “a supplemental instruction is deemed necessary and provided that the ... instruction has been given prior to the time the jury has retired.”
Id.
Collins agrees that the dynamite instruction given in this case comports with the model instruction in
Silvern,
but he argues that the district court committed reversible error by issuing the dynamite instruction without first having included the instruction in the initial jury charge before deliberations commenced. Indeed,
United States v. Brown,
634 F.2d 1069, 1070 (7th Cir.1980), holds that the trial court must include the
Silvern
instruction in its initial charge to the jury, as the district court here did not.
However, Collins waived this objection to the timing of the
Silvern
instruction in most unmistakable and unequivocal terms. The court and both opposing parties were painfully aware of the absence of the
Silvern
instruction in the court’s initial jury charge and discussed whether to give the supplemental instruction after the jury deadlocked. Anxious to reach a verdict and avoid retrial, both parties consented on the record to the issuance of the
Silvern
instruction. Collins attorney explained, “I believe that the law is and still is in the Seventh Circuit that it’s error to give that instruction after the jury has been instructed if it wasn’t tendered originally. We would waive any objection of to the giving of that instruction. I’ve discussed this with Mr. Collins and we would certainly put our waiver on the record.” The court then thrice asked Collins directly whether he consented to the
Silvern
instruction, and each time Collins responded, “Yes, sir.” A claim now regarding the timing of the Silvern instruction is therefore not merely forfeited, but affirmatively waived. As a result, this issue is not subject to appellate review at all because “there is technically no ‘error’ to correct.”
United States v. Ross,
77 F.3d 1525, 1542 (7th Cir.1996);
see also United States v. Olano,
507 U.S. 725, 733, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993).
In
Silvern,
we mentioned that we were exercising our “supervisory power” over the district courts of this Circuit,
Silvern,
484 F.2d at 882, but this does not avert our conclusion. Collins proposes that the
Silvern
timing rule was a supervisory directive to the district courts and cannot be waived by defendants. Collins cites no authority for his ambitious claim, and we have found none. As
Silvern
itself notes, a primary concern driving anxiety over dynamite charges is the due process rights of the parties to a fair trial.
Silvern,
484 F.2d at 882 (explaining in that case “neither the wording of the supplemental charge ... nor the timing and circumstances surrounding the charge violated defendant’s constitutional rights”). When the parties definitively express their consent to a dynamite charge, despite the fact that it had not been included in the initial jury charge, no due process concerns inhere and the district court may find waiver and issue the instruction.
See United States v. Clayton,
172 F.3d 347, 351-52 (5th Cir.1999) (enforcing forfeiture of objections to an
Allen
charge);
United States v. Barbioni,
62 F.3d 5, 7 n. 2 (1st Cir.1995);
United States v. Nguyen,
28 F.3d 477, 484 (5th Cir.1994);
United States v. Meuli,
8 F.3d 1481, 1487 (10th Cir.1993). Any deviation from
Silvern
is not necessarily reversible error. For example, in
United States v. Sblendorio,
830 F.2d 1382, 1388 (7th Cir.1987), the district court gave a dynamite charge that strayed from the model instruction dictated by
Silvern,
but we noted that the defendant did not object. We evaluated the court’s decision for plain error and found none, even though the language of
“Silvern
does not admit of
exceptions.”
Id.
True, we invoked our supervisory authority in
Silvern
to set a bright-line rule with an eye toward avoiding redundant and time-consuming appeals to consider every minute permutation of the
Silvern
instruction.
See Sblendorio,
830 F.2d at 1388. However, that concern is patently absent in cases like this in which the defendant affirmatively waived his objections to the timing of the
Silvern
instruction.
C. Witness Bolstering
Over Collins’s objections, the government introduced into evidence cooperation and plea agreements for ten government witnesses including Allen, Mabry, Hiland, Peters, Logan and Redes. All but one agreement stipulated that the witness agreed to provide “complete and truthful” information and trial testimony, and to reinforce this point, the government elicited testimony from each witness confirming that he or she was obligated by the agreement to be truthful at trial. Collins argues that the district court erred by admitting the agreements into evidence because it impermissibly enhanced the credibility of the witnesses in the jury’s eyes. The admission of the plea agreements was an evidentiary decision, which we review for abuse of discretion.
See United States v. Lewis,
110 F.3d 417, 422 (7th Cir.1997).
Although the government may present evidence about its witnesses’ plea agreements and their concomitant obligations to be truthful, the district court should not permit the government “unnecessarily repetitive references to truthfulness.” Le
wis,
110 F.3d at 421. However, we held in
United States v. Thornton,
197 F.3d 241, 252 (7th Cir.1999), that it was not an abuse of discretion to admit plea agreements containing even more numerous references to truthfulness than in the agreements admitted here. The agreements in
Thornton
required “full, complete and truthful information and testimony” from the witnesses, reserving the government’s right to withdraw the plea agreement and prosecute for perjury if the witness testified falsely.
Id.
at 251. We found that the agreements, plus proffer letters with similar terms, “merely laid out the terms and conditions of the agreements” and “the jury’s role as independent fact finder was not undermined.”
Id.
at 252;
see also Lewis,
110 F.3d at 421;
United States v. Renteria,
106 F.3d 765, 767 (7th Cir.1997). Admission of the plea agreements for the government’s witnesses in this case was not an abuse of discretion, and as in
Thornton,
the court’s cautionary jury instructions removed potential prejudice by directing the jury to consider the government witnesses’ testimony “with caution and great care.”
Thornton,
197 F.3d at 252 n. 4;
United States v. Robbins,
197 F.3d 829, 842 (7th Cir.1999). Remember, however, that
Thornton
reproved prosecutors for coming “perilously close to being unnecessarily repetitive” and to “think twice before risking reversal.”
Thornton,
197 F.3d at 252-53. Collins’s trial preceded
Thornton’s
admonitions, and the government promised at oral argument that it had modified its trial
modus operandi
since
Thornton.
We expect to see appropriate adjustment to the government’s practice in subsequent cases.
In addition, Collins argues that the government impermissibly vouched for the credibility of its witnesses during opening statements and closing argument by referring to the plea agreement requirement of complete truthfulness. A prosecutor may not vouch for the credibility of a witness by either expressing a personal belief in the truthfulness of a witness or implying that facts not before the jury lend credibility to a witness.
See Thornton,
197 F.3d at 252. The government did not commit either form of impermissible vouching in this case because it declared only that the government witnesses had entered plea agreements requiring truthfulness and that the jury ought to consider that fact when evaluating their testimony. As in
United States v. Renteria,
106 F.3d at 767, the plea agreements were in evi
dence and the government “was free to invite the jury to draw a particular inference from this evidence.” Indeed, Collins did not object to these references to the plea agreements, and it was not error for the district court to permit them.
D. Mandatory Life Sentence
Collins challenges the constitutionality of the mandatory life sentence imposed by the “threestrikes” provision in 21 U.S.C. § 841, and we review these challenges
de novo. See United States v. Black,
125 F.3d 454, 458 (7th Cir.1997). Collins, however, concedes that his arguments under the Equal Protection, Double Jeopardy, Due Process and Cruel and Unusual Punishment Clauses were rejected in
United States v. Wicks,
132 F.3d 383, 387-90 (7th Cir.1997), and
United States v. Washington,
109 F.3d 335, 337-38 (7th Cir.1997).
Washington
also addressed and rejected Collins’s argument that the three-strikes mandatory life sentence violates the Ex Post Facto Clause.
Id.
at 338.
Carmell v. Texas,
529 U.S. 513, 120 S.Ct. 1620, 146 L.Ed.2d 577 (2000), does not derogate
Washington. Carmell
overturned under the Ex Post Facto Clause a criminal statute that reduced the quantum of trial evidence required to convict the defendant for sexual assault from the higher level of evidence required by law when his offense was committed.
Carmell,
529 U.S. at -, 120 S.Ct. at 1643. Collins makes a different claim, one squarely rejected in
Washington,
109 F.3d at 338, and distinguished in
Carmell,
529 U.S. at -, 120 S.Ct. at 1627 (differentiating ex post facto laws that change punishment from ex post facto laws that change the legal rules of evidence). Like the defendant in
Washington,
he argues that the three-strikes provision violates the Ex Post Facto Clause because his previous two drug offenses occurred before the enactment of 21 U.S.C. § 841. We see no reason to overrule
Washington
now.
III. CoNclusion
For the foregoing reasons, we Affirm the decisions of the district court.